Numerical regularization of electromagnetic quantum fluctuations in inhomogeneous dielectric media

Abstract

Electromagnetic Casimir stresses are of relevance to many technologies based on mesoscopic devices such as microelectromechanical systems embedded in dielectric media, Casimir induced friction in nanomachinery, microfluidics, and molecular electronics. Computation of such stresses based on cavity QED generally requires numerical analysis based on a regularization process. The scheme described below has the potential for wide applicability to systems involving realistic inhomogeneous media. From a knowledge of the spectrum of the stationary modes of the electromagnetic field the scheme is illustrated by estimating numerically the Casimir stress on opposite faces of a pair of perfectly conducting planes separated by a vacuum and the change in this result when the region between the plates is filled with an incompressible inhomogeneous nondispersive dielectric.

Divergence of Casimir stress in inhomogeneous media

“…This said, two calculations of Casimir forces in inhomogeneous media have recently been presented [13,14]. In [13] it seems the authors do not yet extract numerical results from their formalism.…”

“…An explicit construction of a regularization procedure that goes beyond that used by Lifshitz has apparently not yet been given, and we suspect that it is not straightforward to do so. Meanwhile, in [14], the concern is with the Casimir force between two mirrors with an inhomogeneous medium sandwiched between them. Here the force is calculated, not via Lifshitz theory but from mode summation, obtaining finite results using techniques similar to ζ -function regularization.…”

Cutoff dependence of the Casimir force within an inhomogeneous medium

“…These divergent terms are not canceled by including the contributions to the energy from the other side of the piston, as they were in Eq. (9). This means that in the limit ξ → 0, the force on the piston is discontinuous as a function of α, being finite when α = 0, and infinite as α is moved away from zero.…”

“…The theory has been made quantitative, applying to media described by (ω) and μ(ω) satisfying the Kramers-Kronig relations [2][3][4][5], and Casimir forces have been calculated for a variety of systems and geometries (see, e.g., [6,7]). The purpose of the following discussion is to demonstrate that there seems to be a problem within the theory of the Casimir effect for the case of inhomogeneous media, in spite of recent efforts to solve or circumvent it [8,9]. In a separate paper [10] we have shown that this problem is inherent within the Lifshitz theory of the Casimir effect, where dispersion and dissipation are properly included; the Casimir stress in such cases appears to be infinite, and resists regularization.…”

“…It was recently proposed [9] that one should calculate the Casimir force in an inhomogeneous medium through forming a Laurent expansion of the energy (or pressure) in powers of the regularizing parameter (here ξ ). The regularized quantity is then defined to be the term c 0 in the Laurent expansion that is independent of the regularizing parameter.…”